3 Answers
3

As @GershonGold mentioned in his answer, Mincha time of Shabbat is associated with the passing of Moshe, Yosef, and David. One of the ways we commemorate it is by not greeting someone by wishing them a Good Shabbat at Mincha time. [The Mishna Berurah S"K 6 on Shulchan Aruch (Orach Chaim 292:2) tells us that this is why we say Tzidkatcha after the Amidah in Mincha on Shabbat]

The Rema, in Shulchan Aruch Orach Chaim 292:2, says that for this reason the Ashkenazic custom is not to learn between Mincha and Maariv on Shabbat afternoon. The Mishna Berura (S"K 8) says that this only applies when Mincha is prayed close to the end of the day, otherwise it is not a problem. (He then goes on to describe the limitation of this practice, and the fact that we don't follow it these days).

From there we see that the time of mourning (which would prevent us from saying "Good Shabbat" to one another) is after Mincha, close to darkness.

All this is even more clear if we look at the other reason the Mishna Berurah (S"K 6) brings for saying Tzidkatcha by Mincha. We are saying Tzidduk HaDin (admitting the righteousness of G-d's Judgment) about the wicked people, who are going back to the torments of Gehenom when Shabbat ends, after there reprieve from the pain on Shabbat.

If so, it makes sense that the time of Judgment is close it when it gets dark and Shabbat will soon be over.

Minhag Yisroel Torah Orach Chaim 292:6 "the Minhag is that on Shabbos at the time of Mincha not to wish your friends Shabbos Shalom since it is the time of the passing of Moshe Rabeinu".

Nitei Gavriel Hilchos Yom Tov 2 Page 389:10 mentions in the name of Sefer Minhagei Yeshurin Os 80 and Sefer Matamim, since the passing of Moshe Rabeinu, Yosef HaTzadik, and David HaMelech all occurred at the time of Mincha therefore we do not say Good Shabbos at Mincha.

Although I do not have a source as to when this applies, I asked a fellow who is stringent in this Minhag and he indicated that it starts towards the end of the day - close to sunset.

He was asked the question if there is a source for the minhag that people don't say "Good Shabbos" after Minchas based the the medreshim that Moshe, David and Yosef were nifter during that time. He responds that "ain zeh minhag makubal" ("This is not an accepted minhag") and says to see the end of OC Siman 292 (over there the Rema brings down the Minhag of not learning on Shabbos between Mincha and Maariv. This is based on a Teshuvas HaGaonim that there was such a minhag to not learn during that time because of the "kavod of Moshe" that he was nifter then. There is much to say about this v'ain can makom larich.